This paper addresses the optimal control problem for a dynamic object performing planar motion with constant thrust magnitude. By the end of the maneuver, the object must reach a specified straight-line trajectory with the maximum possible longitudinal velocity. At every moment in time, the object must remain within a specified safe altitude, and a modified linear tangent law is used to satisfy this constraint. The set of initial positions that allow for a solution to the optimal control problem is determined. The paper verifies the sufficient condition for optimality using the Hamilton–Jacobi–Bellman (HJB) equation. Based on this equation, a control algorithm in the form of a synthesis is developed.
Keywords: optimal control, phase constraint, modified linear tangent law, Hamilton–Jacobi–Bellman equation
Received October 5, 2025
Revised January 24, 2026
Accepted February 2, 2026
Funding Agency: This work was supported by the Russian Science Foundation (project no. 23-11-00128, https://rscf.ru/project/23-11-00128/).
Sergey Aleksandrovich Reshmin, Dr. Phys.-Math. Sci., Corresponding Member of RAS, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, 119526 Russia, e-mail: reshmin@ipmnet.ru
Anton Vyacheslavovich Artsibasov, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, 119526 Russia; e-mail: artsibasov@ipmnet.ru
REFERENCES
1. Perkins F.M. Derivation of linear-tangent steering laws. Air Force Report No. SSD-TR66-211, Nov. 1966, 22 p. https://doi.org/10.21236/ad0643209
2. Okhotsimsky D.E., Eneev T.M. Some variational problems related to the launch of the artificial satellite. Uspekhi Fiz. Nauk, 1957, vol. 63, no. 1, pp. 5–32 (in Russian). https://doi.org/10.3367/UFNr.0063.195709b.0005
3. Bryson A.E., Ho Y.-C. Applied optimal control: optimization, estimation, and control. Waltham, Mass., Blaisdell Pub. Co., 1969, 481 p. Translated to Russian under the title Prikladnaya teoriya optimal’nogo upravleniya, Moscow, Mir Publ., 1972, 544 p.
4. Afanas’ev V.N., Kolmanovskii V.B., Nosov V.R. Mathematical theory of control systems design. Transl. From 1st Russian ed. Dordrecht, Springer, 1996, 672 p. https://doi.org/10.1007/978-94-017-2203-2 . Original Russian text (3rd ed.) published in Afanas’ev V. N., Kolmanovskii V. B., Nosov V. R. Matematicheskaya teoriya konstruirovaniya sistem upravleniya, Moscow, Vysshaya Shkola, 2003, 614 p. ISBN: 5-06-004162-X .
5. Reshmin S.A., Bektybaeva M.T. Control of acceleration of a dynamic object by the modified linear tangent law in the presence of a state constraint. Proc. Steklov Inst. Math., 2024, vol. 325, no. 1, pp. S168–S178. https://doi.org/10.1134/S0081543824030131
6. Kim K.S., Park J.W., Tahk M.J., Choi H.L. A PEG-based ascending guidance algorithm for ramjet-powered vehicles. In: Proc. 28th Inter. Congress of the Aeronautical Sciences (ISAC 2012), 2012, 7 p.
7. Hong S.M., Tahk M.-J. Stage optimization of multi-stage anti-air mnoile using co-evolutionary augmented Lagrangian method. In: 2017 25th Mediterranean Conf. Control and Automation (MED). Valletta, Malta, 2017, pp. 1257–1262. https://doi.org/10.1109/MED.2017.7984290
8. Reshmin S.A., Bektybaeva M.T. Accounting for phase limitations during intense acceleration of a mobile robot and its motion in drift mode. Dokl. Math., 2024, vol. 109, no. 1, pp. 38–46. https://doi.org/10.1134/S1064562424701709
9. Roitenberg Ya.N. Avtomaticheskoe upravlenie [Automatic Control]. Moscow, Nauka, 1971, 396 p.
10. Boltyanskii V.G. Sufficient conditions for optimality and a proof of the method of dynamic programming. Izv. Akad. Nauk SSSR. Ser. Mat., 1964, vol. 28, no. 3, pp. 481–514.
11. Fleming W.H., Rishel R.W. Deterministic and stochastic optimal control. NY, Springer, 1975, 222 p. https://doi.org/10.1007/978-1-4612-6380-7
12. Subbotin A.I. Generalized solutions of first order PDEs: the dynamical optimization perspective. Boston, Birkhäuser, 1995, 314 p. https://doi.org/10.1007/978-1-4612-0847-1 . Translated to Russian under the title Obobshchennyye resheniya uravneniy v chastnykh proizvodnykh pervogo poryadka: perspektivy dinamicheskoy optimizatsii, Moscow, Izhevsk, In-t Komp’yut. Issled., 2003, 336 p. ISBN: 5-93972-206-7 .
13. Subbotina N.N., Novoselova N.G. Optimal result in a control problem with piecewise monotone dynamics. Tr. Inst. Mat. Mekh. UrO RAN, 2017, vol. 23, no. 4, pp. 265–280 (in Russian). https://doi.org/10.21538/0134-4889-2017-23-4-265-280
14. Bagno A.L., Tarasyev A.M. Numerical methods for construction of value functions in optimal control problems on an infinite horizon. Izv. Inst. Mat. Inf. Udmurt. Gos. Univer., 2019, vol. 53, pp. 15–26 (in Russian). https://doi.org/10.20537/2226-3594-2019-53-02
15. Rozonoer L.I. Pontryagin’s maximum principle in the theory of optimal systems. I. Avtomat. i Telemekh., 1959, vol. 20, no. 10, pp. 1320–1334 (in Russian).
16. Reshmin S.A., Bektybaeva M.T. Accounting for phase constraints during intensive acceleration by modified linear tangent law. Mech. Solids, 2024, vol. 59, no. 8, pp. 3913–3928. https://doi.org/10.1134/S0025654424607043
Cite this article as: S.A. Reshmin, A.V. Artsibasov. Modified linear tangent law in synthesis form for accounting for a phase constraint and disturbance. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 206–223.
